Problem: Lena is playing her accordion by compressing and stretching it. The accordion's length (in $\text{cm}$ ) $t$ seconds after it's completely stretched is modeled by $A(t)$. Here, $t$ is entered in radians. $A(t) = {25}\cos\left(\pi t\right) + 65$ How long does it take Lena to compress the accordion completely and then stretch it back out to a length of $50\text{ cm}$ ? Round your final answer to the nearest tenth of a second.
Converting the problem into mathematical terms $A(t) = {25}\cos\left({{\pi}}t\right) +65$ has a period of $\dfrac{2\pi}{{\pi}}=2$ seconds. When the accordion is compressed completely, it already passes the $50\text{ cm}$ mark once. Therefore, we want to find the second solution to the equation $A(t)=50$ within the period $0<t<2$. The answer The equation's two solutions within the desired period (rounded to the nearest tenth of a second) are $0.7$ and $1.3$. Therefore, it takes about $1.3$ seconds for Lena to completely compress the accordion and then stretch it back out to a length of $50\text{ cm}$.